{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "initial_id",
   "metadata": {
    "collapsed": true,
    "ExecuteTime": {
     "end_time": "2025-02-28T04:16:58.615605Z",
     "start_time": "2025-02-28T04:16:56.011963Z"
    }
   },
   "outputs": [],
   "source": [
    "import math\n",
    "import torch\n",
    "import d2l"
   ]
  },
  {
   "cell_type": "code",
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 50, x1: -0.171513, x2: 0.090780\n"
     ]
    },
    {
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x1, x2):  # Objective function\n",
    "    return x1 ** 2 + 2 * x2 ** 2\n",
    "\n",
    "def f_grad(x1, x2):  # Gradient of the objective function\n",
    "    return 2 * x1, 4 * x2\n",
    "\n",
    "def sgd(x1, x2, s1, s2, f_grad):\n",
    "    g1, g2 = f_grad(x1, x2)\n",
    "    # Simulate noisy gradient\n",
    "    g1 += torch.normal(0.0, 1, (1,)).item()\n",
    "    g2 += torch.normal(0.0, 1, (1,)).item()\n",
    "    eta_t = eta * lr()\n",
    "    return (x1 - eta_t * g1, x2 - eta_t * g2, 0, 0)\n",
    "\n",
    "def constant_lr():\n",
    "    return 1\n",
    "\n",
    "eta = 0.1\n",
    "lr = constant_lr  # Constant learning rate\n",
    "d2l.show_trace_2d(f, d2l.train_2d(sgd, steps=50, f_grad=f_grad))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2025-02-28T04:16:59.626956Z",
     "start_time": "2025-02-28T04:16:59.486551Z"
    }
   },
   "id": "3fec12505275e982",
   "execution_count": 3
  },
  {
   "cell_type": "markdown",
   "source": [
    "# 12.4.2. Dynamic Learning Rate\n",
    "$\\begin{split}\\begin{aligned}\n",
    "    \\eta(t) & = \\eta_i \\textrm{ if } t_i \\leq t \\leq t_{i+1}  && \\textrm{piecewise constant} \\\\\n",
    "    \\eta(t) & = \\eta_0 \\cdot e^{-\\lambda t} && \\textrm{exponential decay} \\\\\n",
    "    \\eta(t) & = \\eta_0 \\cdot (\\beta t + 1)^{-\\alpha} && \\textrm{polynomial decay}\n",
    "\\end{aligned}\\end{split}$"
   ],
   "metadata": {
    "collapsed": false
   },
   "id": "e8909d9d98f74c9c"
  },
  {
   "cell_type": "code",
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1000, x1: -0.885009, x2: 0.031155\n"
     ]
    },
    {
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   "source": [
    "def exponential_lr():\n",
    "    # Global variable that is defined outside this function and updated inside\n",
    "    global t\n",
    "    t += 1\n",
    "    return math.exp(-0.1 * t)\n",
    "\n",
    "t = 1\n",
    "lr = exponential_lr\n",
    "d2l.show_trace_2d(f, d2l.train_2d(sgd, steps=1000, f_grad=f_grad))"
   ],
   "metadata": {
    "collapsed": false,
    "ExecuteTime": {
     "end_time": "2025-02-28T05:26:25.392343Z",
     "start_time": "2025-02-28T05:26:25.253623Z"
    }
   },
   "id": "6db9d9e11f8cb2e1",
   "execution_count": 6
  },
  {
   "cell_type": "code",
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 50, x1: -0.036251, x2: -0.101176\n"
     ]
    },
    {
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    "def polynomial_lr():\n",
    "    # Global variable that is defined outside this function and updated inside\n",
    "    global t\n",
    "    t += 1\n",
    "    return (1 + 0.1 * t) ** (-0.5)\n",
    "\n",
    "t = 1\n",
    "lr = polynomial_lr\n",
    "d2l.show_trace_2d(f, d2l.train_2d(sgd, steps=50, f_grad=f_grad))"
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